"còn" = still

Let me clarify it a bit more : when you say " Tôi độc thân ", people might think : 1. "you haven't get married until now", 2. "you got married and now you divorced", 3 "you are widowed". So to make it more clear for people to understand what you mean, you say "Tôi còn độc thân"/ "Tôi vẫn còn độc thân" for "I'm still single" "còn/vẫn còn = still"

That means " I am still single "